Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 19G / i

Table of contents
- Solution i

Solution

The irreducible characters are an orthonormal basis of the space of complex class functions for the inner product

⟨ f, h ⟩ = \frac{1}{∣ G ∣} \sum_{g \in G} \overline{f (g)} h (g) .

Thus the row orthogonality relation and completeness statement are

\frac{1}{∣ G ∣} g \in G \sum \overline{χ_{i} (g)} χ_{k} (g) = δ_{ik}, f = i \sum ⟨ χ_{i}, f ⟩ χ_{i} .

In particular the number of irreducible characters equals the number of conjugacy classes. In class-representative notation, the orthogonality relation is

\sum_{j} \overline{χ_{i} (g_{j})} χ_{k} (g_{j}) /∣ Z (g_{j}) ∣ = δ_{ik}

.

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